Stochastic differential game formulation on the reinsurance and investment problem

Abstract

This paper focuses on a stochastic differential game between two insurance companies, a big one and a small one. The big company has sufficient asset to invest in a risk-free asset and a risky asset and is allowed to purchase proportional reinsurance or acquire new business, and the small company can transfer part of the risk to a reinsurer via proportional reinsurance. The game studied here is zero-sum, where the big company is trying to maximise the expected exponential utility of the difference between two insurance companies’ surpluses at the terminal time to keep its advantage on surplus, while simultaneously the small company is trying to minimise the same quantity to reduce its disadvantage. Particularly, the relationships between the surplus processes and the price process of the risky asset are considered. By applying stochastic control theory, we provide and prove the verification theorem and obtain the Nash equilibrium strategy of the game, explicitly. Furthermore, numerical simulations are presented to illustrate the effects of parameters on the equilibrium strategy as well as the economic meanings behind.

Keywords:

• reinsurance and investment
• stochastic differential game
• Nash equilibrium strategy
• exponential utility maximisation
• stochastic control

Acknowledgements

The authors would be very grateful to referees for their suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11201335], [grant number 11301376].